We now return to some analytic aspects of Joyce’s proposal [41] related to the Lagrangian mean curvature flow (LMCF) inside a Calabi-Yau manifold. Recall a smooth mean curvature flow means a family of immersions $\iota_{t}:L\to M$ parametrised by time $t\in[0,T)$, such that the velocity is equal to the mean curvature:

$$\partial_{t}\vec{x}=\vec{H}.$$

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The starting point of LMCF is an early observation of Smoczyk, which justifies the name:

Assuming $(L_{t})$ is a smooth and compact LMCF inside a Calabi-Yau ambient manifold, then if the initial Lagrangian is graded, i.e. the Lagrangian angle is well defined as a real valued function on the Lagrangian, then so is $L_{t}$. The grading is highly desirable, because of the foundational fact that the Lagrangian angle function $L_{t}\to\mathbb{R}$ satisfies the heat equation

$$(\partial_{t}-\Delta_{L_{t}})\theta_{t}=0,$$

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which among many other things, implies that $\sup_{L_{t}}\theta_{t}$ can only decrease in time, and $\inf_{L_{t}}\theta_{t}$ can only increase in time, and in particular the almost calibrated condition would be preserved by the flow. Inside a Calabi-Yau manifold, the mean curvature of $L_{t}$ is related to the Lagrangian angle $\theta_{t}$ by an appealing formula:

$$\vec{H}=J\nabla\theta_{t},$$

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where $\nabla\theta$ stands for the gradient of $\theta$ along $L_{t}$. Along the LMCF

$$\omega(\partial_{t}\vec{x},\cdot)=\omega(\vec{H},\cdot)=\omega(J\nabla\theta_{t},\cdot)=-d\theta_{t},$$

so the Lagrangians evolve by the local Hamiltonian function $-\theta_{t}$ up to an additive constant.

Mean curvature flow in codimension greater than one does not satisfy the avoidance principle. As such embedded Lagrangians can become immersed during the flow, so the program should at least include immersed Lagrangians. Joyce further suggests that certain ‘stable Lagrangian singularities’ should be admitted. For instance, inside Calabi-Yau 3-folds one should allow Lagrangians with local conical singularity modelled on the Harvey-Lawson $T^{2}$-cone [41, Example 2.7]. The adjective ‘stable’ here means that the flow should preserve this class of singularities at least for a short amount of time, even if one makes a generic perturbation of the initial data.